Topological embeddings into random 2-complexes

Michael Farber, Tahl Nowik*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We consider 2-dimensional random simplicial complexes Y in the multiparameter model. We establish the multiparameter threshold for the property that every 2-dimensional simplicial complex S admits a topological embedding into Y asymptotically almost surely. Namely, if in the procedure of the multiparameter model on n vertices, each i-dimensional simplex is taken with probability pi = pi(n), then the threshold is (Formula presented.). Our claim in one direction is in fact slightly stronger, namely, we show that if (Formula presented.) is sufficiently larger than (Formula presented.) then every S has a fixed subdivision S′ which admits a simplicial embedding into Y asymptotically almost surely. In the other direction we show that if (Formula presented.) is sufficiently smaller than (Formula presented.), then asymptotically almost surely, the torus does not admit a topological embedding into Y.

Original languageEnglish
Pages (from-to)664-675
Number of pages12
JournalRandom Structures and Algorithms
Issue number4
StatePublished - Jul 2021
Externally publishedYes


  • multiparameter threshold
  • random simplicial complexes


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